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1、统计物理学习讲义统计物理学习讲义中科院数学院复杂系统研究中心复杂系统学习班 (CSSGBJ)韩 靖2003年10月27日统计物理、自旋玻璃和复杂系统统计物理、自旋玻璃和复杂系统统计物理做什么?统计物理做什么?自旋玻璃自旋玻璃(Spin Glasses)是什么?是什么?它们在复杂系统研究中有何应用?它们在复杂系统研究中有何应用?它们的局限性?它们的局限性?探讨:对我们的研究有何启发?探讨:对我们的研究有何启发?学习提纲和计划学习提纲和计划 (欢迎补充修改欢迎补充修改)基本概念介绍nEntropy, Boltzmann分布(partition function)nExample: K-SAT问题的
2、相变Dynamics and Landscapesn各态历尽, landscapes, Monte Carlo SimulationnExample: Simulated Annealing(模拟退火)Meanfield, Replica Symmetry, Cavity MethodsnMeanfield 用于网络动力学的例子nReplica Symmetry 用于组合问题的例子nCavity Methods: Survey Propagation Critical Phenomena & Power-lawn相变nSOC, HOT/COLD理论谁报名来主讲?统计物理统计物理Statisti
3、cal physics is about systems composed of many parts. 集体行为集体行为 组合数学和概率理论组合数学和概率理论Traditional examples:n气体、液体、固体 - 原子或分子;n金属、半导体 - 电子;n量子场 - 量子,电磁场 - 光子等Complex systems examples:n生态系统 - 物种n社会系统 - 人n计算机网络 - 计算机n市场 - 经纪人agentn鱼群 - 鱼、鸟群 - 鸟、蚁群 - 蚂蚁n组合问题组合问题 变量变量 研究复杂系统为什么要学习统计物理研究复杂系统为什么要学习统计物理?Collectiv
4、e Behavior 群群体行为体行为集体行为:n系统由大量相似的个体组成n全局行为不依赖于个体的精确细节,而相互作用必须合理定义,并且不要太复杂;n个体在单独存在的行为与在整体中的行为很不一样.(在整体中各个体行为变得相似);n相互作用的类型:吸引、抗拒、对齐n主要的集体现象:相变、模式形成、群组运动、同步 n研究手段:统计物理、多主体计算机模拟“磁化”现象:go个体行为 邻居动作的平均方向同步掌声恐慌现象http:/angel.elte.hu/vicsek/自旋玻璃自旋玻璃(Spin Glasses)简单的理想模型,性质丰富,易于研究个体:spin si; 系统:多个spin局部相互作用以
5、最简单的Ising模型为例:nsi=1 或者 1n在lattice上排列,相邻spin之间有相互作用n能量(Hamiltonian):E = - J(i-1)isi-1siJij0, 偏好相邻同向;Jij0, 偏好相邻不同向;Jij=0,无相互作用考虑外部场 E = - Jijsisj - hisi性质:有序/无序、受挫、相变、对称破缺现实中的例子:组合问题、恐慌人群、经济模型(-)(+)(+) ?sisi+1si-1J(i-1)iJi(i+1)E=- JijsisjSpin GlassConfiguration r = s1,s2,snHamiltonian (E, Cost functio
6、n): E(r)J=HJ(r) = -JiksiskQuenched variable: J, random variable a probability distribution P(J)Different Spin model: different P(J)Notation:=PJ(s)g(s)So-called Disorder: Structural parameter J is random and have large complexity自旋玻璃例子自旋玻璃例子- K-SAT问题问题经典NP-完全问题N个布尔变量: xi=True/False, si=1/-1M个clauses:
7、 M个含k个变量的逻辑表达式K=3, 3-SAT: c1:x1 or (not x3) or x8, c2:(not x2) or x3 or (not x4), c3:x3 or x7 or x9,目标:满足所有M个clauses 的 N个布尔变量的一组赋值Spin glass 的能量 E =- a=1,M(Ca =T),Ground State E=-M 解状态结果:当K=3, M/N 4.25, 问题求解困难 恐慌现象恐慌现象行人建模:期望移动速度、与他人的排斥力、与墙壁的作用力、个人速度的扰动恐慌(由于火灾或者大众心理):n人们希望移动更快n人与人之间的物理冲突更厉害;n出口处障碍、堵
8、塞形成;n危险压力出现;n人群开始出现大众恐慌心理;n看不到其它的出口;计算机模拟实验: (Go) n单出口房间:无恐慌、恐慌、惊跑、带圆柱、火灾n走廊:直走廊、中间加宽的走廊n人群:个人主义、群体心理、两者综合Begin统计物理能做什么?怎么做?基本点:n只关心状态的概率,并不关心演化的过程(假设各态历经)n熵最大核心: Boltzmann分布 (partition function)学习提纲和计划学习提纲和计划基本概念介绍nEntropy, Boltzmann分布(partition function)nExample: K-SAT问题的相变Dynamics and Landscapesn
9、各态历尽, landscapes, Monte Carlo SimulationnExample: Simulated Annealing(模拟退火)Meanfield, Replica Symmetry, Cavity MethodsnMeanfield 用于网络动力学的例子nReplica Symmetry 用于组合问题的例子nCavity Methods: Survey Propagation Critical Phenomena & Power-lawn相变nSOC, HOT/COLD理论EntropyMicrostate r: a specific configuration of
10、systemMacrostate R: an evaluation value(R): number of microstates related to a macrostateMicro-canonical entropy: S(R)=k log (R) More General forms:A macrostate R: pi for system be found in a microstate i A distribution of microstates.Gibbs Entropy: S(R) =-k pi logpi Maximum the most possible distri
11、bution of microstates Without constraint on pi, pi=1/N S is maximized (n ni i)=M!/n)=M!/n1 1!n!n2 2!.!.n nNN!, !, p pi i= =n ni i/M/MWith Constraint on pi: Partition Function ZObservable quantity E (Hamiltonian)Ergodic Hypothesis (time average=ensemble average)We know: nFrom experiments: , nEi for a
12、ll ri, and = = piEi, pi=1.We want to know the most probable distribution of microstates Maximize S=-kpilogpi and we get: pi=e-Ei/Z, Z=ie-Ei (=(kT)-1)So, pi and is decided by Ei and Knowing or T and Ei, we can define the most possible distribution of microstates pi and Z T Z distribution is less symm
13、etricalToy ExampleThree microstates: E1=0, E2=2, E3=3We have p1E1+p2E2+p3E3= e.g. 2p2+3p3=, and p1+p2+p3=1 3 temperatures: decreasing order of TZ p1p2p311.50.105 2.540 0.393 0.319 0.287210.420 1.716 0.583 0.252 0.16530.31.083 1.154 0.867 0.099 0.034Important conceptslPartition function: Z(T,E)=re- E
14、(r)/T Knowing this, we can do a lot of things!Variance of E, #sol, lFree Energy: F = -k T lnZ (?)lEntropy S=- (F/ T)E=-k pilnpi Z and #sol (ground state)Z (T)=re-E(r)/T = H=1,2,r|E(r)=H e-H/T When T0, system are most likely in the ground state. e-E(r)/T 0 except E(r)=0Z(0)= r|E(r)=0 e-0 = r|E(r)=0So
15、, number of ground states = Z(0).In T0, Z also counts other r that E(r)0. But the lower T, the r with lower E(r) Z counts. Z is decreasing when T is decreasing.The K-SAT result considers T=0.学习提纲和计划学习提纲和计划基本概念介绍nEntropy, Boltzmann分布(partition function)nExample: K-SAT问题的相变Dynamics and Landscapesn各态历尽
16、, landscapes, Monte Carlo SimulationnExample: Simulated Annealing(模拟退火)Meanfield, Replica Symmetry, Cavity MethodsnMeanfield 用于网络动力学的例子nReplica Symmetry 用于组合问题的例子nCavity Methods: Survey Propagation Critical Phenomena & Power-lawn相变nSOC, HOT/COLD理论各态历尽各态历尽对任意2个系统状态r1和r2, r1可以经过有限部变换到r2. 00011011熵最大分布
17、的三个条件熵最大分布的三个条件 Rij=probability of ri changes to rj 方程的平衡状态是熵最大分布,必须要满足:np=Rp, R 有唯一的主特征向量(特征值为1)n各态历经n细致平衡:平衡态时,piRij=pjRjiErgodicity breaking and LandscapeMapping of microstates onto energiesbarrierr1r2r3rnVery high, unlikely to cross, when system size is large,T is low:pi/pj=e-(Ei-Ej)/TMonte Carl
18、o Simulation设定状态转换矩阵,使得系统演化服从我们希望的状态分布 P。如果各态历尽和细致平衡,有 把P代入就可以得到Rij Simulated Annealing目标P是Boltzmann分布:pie-Ei/T。Rij/Rji=e-(Ej-Ei)/T Rij= 1if EjEi e-(Ej-Ei)/T if EjEiSimulated Annealing:nWe want to minimize EnT=0, ergodicity breaking, favors minimal EnT0, barriers can be crossed, favors more states M
19、ost problems have many metastable states (local optima), various scales of barriers heights学习提纲和计划学习提纲和计划基本概念介绍nEntropy, Boltzmann分布(partition function)nExample: K-SAT问题的相变Dynamics and Landscapesn各态历尽, landscapes, Monte Carlo SimulationnExample: Simulated Annealing(模拟退火)Meanfield, Replica Symmetry,
20、Cavity MethodsnMeanfield 用于网络动力学的例子nReplica Symmetry 用于组合问题的例子nCavity Methods: Survey Propagation Critical Phenomena & Power-lawn相变nSOC, HOT/COLD理论Replica Approach and P(J)For a given J, free energy density:fJ=-1/(N) ln ZJFor a P(J), we want to know: =P(J)fJFor n replicas: Zn=JP(J)(ZJ)n (ZJ)n=s1s2sn
21、 exp-a=1nHJ(sa)si is the i th replica. fn=-1/(nN) ln Zn, ln Z= Lim n0 (Zn-1/n)We get: = Lim n0 fn f0参考教材参考教材 http:/ Newman 2001 复杂系统暑期学校教材 http:/www.santafe.edu/mark/budapest01/ K-SAT相变: Nature, Vol 400, July 1999, p133-137Survey Propagation: Science, Vol 297, Aug. 2002, p812-815, p784-785.SOC: 大自然如何工作, Per Bak.HOT/COLD:nHOT: Highly Optimized Tolerance: A Mechanism for Power Laws in Designed Systems. J. M. Carlson, John Doyle. (April 27, 1999)nCOLD: Optimal design, robustness, and risk aversion. M. E. J. Newman, Michelle Girvan and J. Doyne Farmer